U.S. patent number 8,155,462 [Application Number 11/968,030] was granted by the patent office on 2012-04-10 for system of master reconstruction schemes for pyramid decomposition.
This patent grant is currently assigned to FastVDO, LLC. Invention is credited to Lijie Liu, Pankaj N. Topiwala, Trac D. Tran.
United States Patent |
8,155,462 |
Tran , et al. |
April 10, 2012 |
System of master reconstruction schemes for pyramid
decomposition
Abstract
A reconstruction system for digital signals processed by the
laplacian pyramid including a master lifting-based parameterization
reconstruction scheme. The system also involves the design of
low-complexity FIR linear-phase integer-coefficient filtering
operators for lapacian pyramid decimation and interpolation stages
that deliver a minimum mean-squared error reconstruction.
Inventors: |
Tran; Trac D. (Columbia,
MD), Liu; Lijie (Skillman, NJ), Topiwala; Pankaj N.
(Clarksville, MD) |
Assignee: |
FastVDO, LLC (Columbia,
MD)
|
Family
ID: |
39641287 |
Appl.
No.: |
11/968,030 |
Filed: |
December 31, 2007 |
Prior Publication Data
|
|
|
|
Document
Identifier |
Publication Date |
|
US 20080175500 A1 |
Jul 24, 2008 |
|
Related U.S. Patent Documents
|
|
|
|
|
|
|
Application
Number |
Filing Date |
Patent Number |
Issue Date |
|
|
60877850 |
Dec 29, 2006 |
|
|
|
|
Current U.S.
Class: |
382/240 |
Current CPC
Class: |
H04N
19/60 (20141101); H04N 19/33 (20141101) |
Current International
Class: |
G06K
9/36 (20060101) |
Field of
Search: |
;382/162,166,232-233,238-240,260 ;708/298,300,308,313 |
References Cited
[Referenced By]
U.S. Patent Documents
Other References
PJ. Burt and E.H. Adelson, "The Laplacian pyramid as a compact
image code," IEEE Trans. Commun., vol. COM-31, pp. 532-540, Apr.
1983. cited by other .
P.P Vaidyanathan, Multirate Systems and Filter Banks, Prentice
Hall, 1993. cited by other .
M. Vetterli and J. Kovacevic, Wavelets and Subband Coding, Prentice
Hall, 1995. cited by other .
Z. Cvetkovic and M. Vetterli, "Oversampled Filter Banks", IEEE
Trans. Signal Processing, vold 46, No. 5, pp. 1245-1255, May 1998.
cited by other .
S. Mallat, A Wavelet Tour of Signal Processing, Second Edition,
Academic Press, 1999. cited by other .
D. Taubman and M. Marcellin, JPEG2000: Image Compression
Fundamentals, Practice and Standards, Kluwer Academic Publishers,
2001. cited by other .
M.N.Do and M. Vetterli, "Frame Pyramid," IEEE Trans. Signal
Processing Mag., 2007. cited by other .
D. Santa-Criz, J. Reichel, and F. Ziliani, "Opening the Laplacian
Pyramid for Video Coding," Proc. ICIP, pp. 672-675 Sep. 2005. cited
by other .
M. Flierl and P. Vanderghyenst, "An Improved Pyramid for Spatially
Scalable Video Coding," in Proc. IEEE International Conference on
Image Processing, Genova, Italy, Sep. 2005, vol. 2, pp. 878-881.
cited by other .
L. Gan and C. Ling, "Computation of the Dual Frame: Forward and
Backward Greville Formulas", Proc. IEEE Int. CASSP, 2007. cited by
other .
L. Liu, L. Gan, and T.D. Tran, "General Reconstruction of Laplacian
Pyramid and its Dual Frame Solutions," 41.sup.st Conf. on Info Sci.
and Sys., Baltimore, MD, Mar. 2007. cited by other .
H. Schwarz, D. Marpe, and T. Wiegand, "Overview of the Scalable
Extension of the H.264/MPEG-4 AVC Video Coding Standard," IEEE
Trans. Circuits Syst. Video Tech., Sep. 2007. cited by other .
J. Kovacevic and A. Chebira, "Life Beyond Bases: The Advent of
Frames," IEEE Signal Processing Mag., 2007. cited by other.
|
Primary Examiner: Couso; Jose
Attorney, Agent or Firm: Dickstein Shapiro LLP
Parent Case Text
CROSS-REFERENCE TO RELATED APPLICATIONS
This application claims priority to provisional U.S. patent
application entitled, High-Performance Low-Complexity Re-Sampling
Filters For Scalable Video Codec, filed Dec. 29, 2006, having a
Ser. No. 60/877,850, the disclosure of which is hereby incorporated
by reference in its entirety. U.S. Pat. No. 6,421,464, entitled
"Fast Lapped Image Transforms Using Lifting Steps," is also hereby
incorporated by reference in its entirety.
Claims
What is claimed is:
1. A computer system with an optimal Laplacian pyramid processing
system (OLaPPS), comprising: a computer configured to store and
manipulate uni- and multi-dimensional discrete digital signals; and
a decimation filter component of a Laplacian pyramid processing
system associated with said computer for processing digital signal
elements selected from a set of dimensions from one or more of the
uni- and multi-dimensional discrete digital signals, the decimation
filter component having a high-resolution signal as an input and a
decimation signal as an output; and an interpolation filter
component of a Laplacian pyramid processing system associated with
said computer for processing digital signal elements selected from
a set of dimensions from one or more of the uni- and
multi-dimensional discrete digital signals, the interpolation
filter component having the decimation signal as an input and a
reconstructed signal as an output, wherein the decimation signal
retains maximum energy and the reconstructed signal has minimum
mean square error relative to the original high resolution
signal.
2. The system of claim 1, wherein filter elements of the decimation
filter component are rational approximations of optimal filter
elements.
3. The system of claim 1, wherein filter elements of the
interpolation filter component are rational approximations of
optimal filter elements.
4. The system of claim 1, wherein the filter elements of the
decimation and interpolation filter components are dyadic rational
approximations of optimal filter elements.
5. A computer system with an enhanced reconstruction stage in an
optimal Laplacian pyramid processing system (OLaPPS), comprising: a
computer configured to store and manipulate uni- and
multi-dimensional discrete digital signals; and a Laplacian pyramid
processing system associated with said computer for processing a
plurality of digital signal elements selected from one or more of
the uni- and multi-dimensional discrete digital signals, the
processing system comprising: a Laplacian pyramid decomposition
stage, including a first decimation having a signal as an input and
a coarse approximation of the signal as an output, and
interpolation having the coarse approximation as an input and whose
output is subtracted from the signal to result in a detail signal;
an intermediate stage; a Laplacian pyramid reconstruction stage,
including having the coarse approximation and detail signal as
inputs, and a reconstructed signal as an output; and an enhanced
reconstruction stage having the coarse approximation and
reconstructed signal as inputs, wherein a second decimation is
applied to the reconstructed signal, the output of said second
decimation is subtracted from the coarse approximation to result in
an intermediary signal to which a prediction is applied to result
in an enhanced reconstructed signal as output.
6. The system of claim 5, wherein the first and second decimation
filter elements are rational.
7. The system of claim 5, wherein the interpolation filter elements
are rational.
8. The system of claim 5, wherein the first and second decimation
filter elements are dyadic rational.
9. The system of claim 5, wherein the interpolation filter elements
are dyadic rational.
10. The system of claim 5, wherein the first decimation filter
retains maximum energy in the coarse approximation signal.
11. The system of claim 5, wherein the interpolation filter is such
that the reconstructed signal is a minimum mean square error
approximation of the original signal.
12. The system of claim 5, wherein the first and second decimation
filter and interpolation filter elements are rational
approximations of optimal filter elements.
13. The system of claim 1 wherein the first or second decimation
filter is a 13-tap low pass filter.
14. The system of claim 5 wherein the interpolation filter has a
filter length selected from one of the group of 3, 4, 5, 6, 7, 8,
10, and 11.
15. A system of claim 1, wherein the first or second interpolation
filter has a filter length selected from one of the group of 3, 4,
5, 6, 7, 8, 10, and 11.
16. A system of claim 1, wherein the first or second decimation
filter has a filter length selected from one of the group of 4, 5,
6, 8, 9, 10, 11, and 17.
17. A system of claim 5, wherein the first or second decimation
filter has a filter length selected from one of the group of 4, 5,
6, 8, 9, 10, 11, and 17.
18. The system of claim 5 wherein the first or second decimation
filter is a 13-tap low pass filter.
19. The system of claim 18, wherein the first and second decimation
filters are close approximations.
20. The system of claim 16, wherein the first and second decimation
filters are close approximations.
21. The system of claim 17, wherein the first and second decimation
filters are close approximations.
22. The system of claim 15, wherein the first and second
interpolation filters are close approximations.
23. The system of claim 14, wherein the interpolation filters are
close approximations.
24. The system of claim 5, wherein operators of the first and
second decimation may be arbitrary.
25. The system of claim 5, wherein operators of the interpolation
may be arbitrary.
26. The system of claim 5, wherein operators of the enhanced
reconstruction stage may be arbitrary.
27. A computer system with one of two components of an optimal
Laplacian pyramid processing system (OLaPPS), comprising: a
computer configured to store and manipulate uni- and
multi-dimensional discrete digital signals; and a decimation filter
component of a Laplacian pyramid processing system associated with
said computer for processing digital signal elements selected from
a set of dimensions from one or more of the uni- and
multi-dimensional discrete digital signals, the decimation filter
component having a high-resolution signal as an input and a
decimation signal as an output, wherein the decimation signal
retains maximum energy.
28. A computer system with one of two components of an optimal
Laplacian pyramid processing system (OLaPPS), comprising: a
computer configured to store and manipulate uni- and
multi-dimensional discrete digital signals; and an interpolation
filter component of a Laplacian pyramid processing system
associated with said computer for processing digital signal
elements selected from a set of dimensions from one or more of the
uni- and multi-dimensional discrete digital signals, the
interpolation filter component having a coarse approximation signal
as an input and a reconstructed signal as an output, wherein the
reconstructed signal has minimum mean square error relative to an
original high resolution signal represented by the coarse
approximation signal.
29. A computer system with an optimal Laplacian pyramid processing
system (OLaPPS), comprising: a computer configured to store and
manipulate uni- and multi-dimensional discrete digital signals; and
a Laplacian pyramid processing system associated with said computer
for processing digital signal elements selected from a set of
dimensions from one or more of the uni- and multi-dimensional
discrete digital signals, the processing system comprising: a
Laplacian pyramid decomposition stage, including a first decimation
having a signal as an input and a coarse approximation of the
signal as an output, and a first interpolation having the coarse
approximation as an input and whose output is subtracted from the
signal to result in a detail signal; an intermediate stage; and a
Laplacian pyramid reconstruction stage, including a second
interpolation having the coarse approximation as an input and whose
output is summed with the detail signal to result in a first
reconstruction stage signal; a second decimation having the first
reconstruction stage signal as an input and whose output is
subtracted from the coarse approximation to result in a second
reconstruction stage signal; and a prediction having the second
reconstruction stage signal as an input and whose output is summed
with the first reconstruction stage signal to result in a
reconstructed signal as an output, wherein the first decimation
retains maximum energy in the coarse approximation and the
reconstructed signal is simultaneously a minimum mean square error
approximation of the original signal.
30. The system of claim 29, wherein filter elements of the first
and second decimations are rational approximations of optimal
filter elements.
31. The system of claim 29, wherein filter elements of the first
and second interpolations are rational approximations of optimal
filter elements.
32. The system of claim 29, wherein the filter elements of the
first and second decimation and first and second interpolation
filters are dyadic rational approximations of optimal filter
elements.
33. An apparatus with one of two jointly-defined components of an
optimal Laplacian pyramid processing system (OLaPPS), comprising: a
signal processing device configured to receive, store, manipulate
and forward uni- and multi-dimensional discrete digital signals;
and a decimation filter component of a Laplacian pyramid processing
system associated with said signal processing device for processing
digital signal elements selected from a set of dimensions from one
or more of the uni- and multi-dimensional discrete digital signals,
the decimation filter component having a high-resolution signal as
an input and a decimation or coarse approximation signal as an
output, wherein the decimation signal retains maximum energy.
34. The apparatus of claim 33, wherein filter elements of the
decimation filter component are rational approximations of optimal
filter elements.
35. The apparatus of claim 34, wherein the filter elements are
dyadic rational approximations of the optimal filter elements.
36. The apparatus of claim 34, wherein the decimation filter
component has an integer length in the range of 5-17 taps.
37. An apparatus with one of two jointly-defined components of an
optimal Laplacian pyramid processing system (OLaPPS), comprising: a
signal processing device configured to receive, store, manipulate
and forward uni- and multi-dimensional discrete digital signals;
and an interpolation filter component of a Laplacian pyramid
processing system associated with said signal processing device for
processing digital signal elements selected from a set of
dimensions from one or more of the uni- and multi-dimensional
discrete digital signals, the interpolation filter component having
a decimation or coarse approximation signal as an input and a
reconstructed signal as an output, wherein the reconstructed signal
has minimum mean square error relative to an original high
resolution signal.
38. The apparatus of claim 37, wherein filter elements of the
interpolation filter component are rational approximations of
optimal filter elements.
39. The apparatus of claim 38, wherein the filter elements are
dyadic rational approximations of the optimal filter elements.
40. The apparatus of claim 38, wherein the interpolation filter
component has an integer length in the range of 5-17 taps.
41. An apparatus with an optimal Laplacian pyramid processing
system (OLaPPS), comprising: a signal processing device configured
to receive, store, manipulate and forward uni- and
multi-dimensional discrete digital signals; and a Laplacian pyramid
processing system associated with said signal processing device for
processing digital signal elements selected from a set of
dimensions from one or more of the uni- and multi-dimensional
discrete digital signals, the processing system comprising: a
Laplacian pyramid decomposition stage, including a first decimation
having a signal as an input and a coarse approximation of the
signal as an output, and a first interpolation having the coarse
approximation as an input and whose output is subtracted from the
signal to result in a detail signal; an intermediate stage; and a
Laplacian pyramid reconstruction stage, including a second
interpolation having the coarse approximation as an input and whose
output is summed with the detail signal to result in a first
reconstruction stage signal; a second decimation having the first
reconstruction stage signal as an input and whose output is
subtracted from the coarse approximation to result in a second
reconstruction stage signal; and a prediction having the second
reconstruction stage signal as an input and whose output is summed
with the first reconstruction stage signal to result in a
reconstructed signal as an output, wherein the first decimation
retains maximum energy in the coarse approximation and the
reconstructed signal is simultaneously a minimum mean square error
approximation of the original signal.
42. The apparatus of claim 41, wherein the filter elements of the
first and second decimation and first and second interpolation
filters are dyadic rational approximations of optimal filter
elements.
Description
FIELD OF THE INVENTION
The present invention relates generally to the processing of uni-
and multi-dimensional discrete signals such as audio, radar, sonar,
natural images, photographs, drawings, multi-spectral images,
volumetric medical image data sets, video sequences, etc, at
multiple resolutions that are captured directly in digital format
or after they have been converted to or expressed in digital
format. More particularly, the present invention relates to the
processing of image/video (visual) data and the use of novel
decomposition and reconstruction methods within the pyramid
representation framework for digital signals that have been
contaminated by noise.
BACKGROUND OF THE INVENTION
Multi-scale and multi-resolution representations of visual signals
such as images and video are central for image processing and
multimedia communications. They closely match the way that the
human visual system processes information, and can easily capture
salient features of signals at various resolutions. Moreover,
multi-resolution algorithms offer computational advantages and
usually have more robust performance. For example, as a scalable
extension of video coding standard H.264/MPEG-4 AVC, the SVC
standard has achieved a significant improvement in coding
efficiency, as well as the degree of scalability relative to the
scalable profiles of previous video coding standards. The basic
structure for supporting the spatial scalability in this new
standard is the well-known Laplacian Pyramid.
The Laplacian Pyramid (hereinafter "LP"), also called Laplace
Pyramid in the current literature, and introduced by P. J. Burt and
E. H. Adelson in 1983, is a fundamental tool in image/video
processing and communication. It is intimately connected with
resampling such that every pair of up sampling and down sampling
filters corresponds to an LP, by computing the detail difference
signal at each step. Vice versa, by throwing away the detail
signal, up- and down-sampling filters result. Traditionally, LPs
have been focused on resamplings of a factor of 2, but the
construction can be generalized to other ratios. In the most
general setting, non-linear operators can be employed to compute
the coarse approximation as well as the detail signals. The LP is
one of the earliest multi-resolution signal decomposition schemes.
It achieves the multi-scale representation of a signal as a coarse
signal at lower resolution together with several detailed signals
as successive higher resolution.
This is demonstrated in FIG. 1 where H(z) 14 is often called the
Decimation Filter and G(z) 16 is often referred to as the
Interpolation Filter. Such a representation is implicitly using
over-sampling. Hence, in compression applications, it is normally
replaced by sub-band coding or the wavelet transform, which are all
critically-sampled decomposition schemes.
The LP is the foundation for spatial scalability in numerous video
coding standards, such as MPEG-2, MPEG-4, and the recent H.264
Scalable Video Coding (SVC) standard propounded in the September
2007 article entitled "Overview of the scalable extension of the
H.264/MPEG-4 AVC video coding standard", by H. Schwarz, D. Marpe,
and T. Wiegand. The LP provides an over-complete representation of
visual signals, which can capture salient features of signals at
various resolutions. It is an implicitly over-sampling system, and
can be characterized as an over-sampled filter bank (hereinafter
"FB") or frame. As the inverse of an over-sampled analysis FB,
beside the conventional reconstruction scheme depicted in FIG. 2,
the LP reconstruction actually has an infinite number of
realizations that can satisfy the perfect reconstruction
(hereinafter "PR") property. Despite the sampling redundancy, the
LP still has its occasional advantages over the critically sampled
wavelet scheme. In the LP, each pyramid level only down-samples the
low-pass channel and generates one band-pass signal. Thus, the
resulting signal does not suffer from the "scrambled" frequencies,
which normally exist in critical sampling scheme because the
high-pass channel is folded back into the low frequency after
sampling. Therefore, the LP enables further decomposition to be
employed on its band-pass signals, generating some
state-of-the-arts multi-resolution image processing and analysis
tools.
The LP decomposition framework provides a redundant representation
and thus has multiple reconstruction methods. Given an LP
representation, the original signal usually can be reconstructed
simply by iteratively interpolating the coarse signal and adding
the detail signals successively up to the final resolution.
However, when the LP coefficients are corrupted with noise, such
reconstruction method can be shown to be suboptimal from a filter
bank point of view. Treating the LP as a frame expansion, M. N. Do
and M. Vetterli proposed in 2003 a frame-based pyramid
reconstruction scheme, which has less error than the usual
reconstruction method. They presented from frame theory a complete
parameterization of all synthesis FBs that can yield PR for a given
LP decomposition with a decimation factor M. Such a general LP
reconstruction has M.sup.2+M free parameters. Moreover, they
revealed that the traditional LP reconstruction is suboptimal, and
proposed an efficient frame-based LP reconstruction scheme.
However, such frame reconstruction approaches require the
approximation filter and interpolation filter to be biorthogonal in
order to achieve perfect reconstruction. Since a biorthogonal
filter can cause significant aliasing in the down-sampled lowpass
subband, it may not be advisable for spatially scalable video
coding.
To keep the same reconstruction scheme but overcome the
bi-orthogonality limitation in the frame-based pyramid
reconstruction, a method called lifted pyramid was presented by M.
Flierl and P. Vandergheynst in 2005 to improve scalable video
coding efficiency. Therein, the lifting steps are introduced into
pyramid decomposition and any filters can be applied to have
perfect reconstruction. The lifted pyramid introduced an additional
lifting step into the LP decomposition so that the perfect
reconstruction condition can be satisfied. where the lifting steps
are introduced into pyramid decomposition and any filters can be
applied to have perfect reconstruction. When compared to the
conventional LP, however, the low-solution representation of the
lifted pyramid has more significant high-frequency components and
requires larger bit rate because of the spatial update step in the
decomposition. Thus, it is undesirable in the context of scalable
video compression.
A similar modified LP scheme called Laplacian Pyramid with Update
(hereinafter "LPU") was presented by D. Santa-Cruz, J. Reichel, and
F. Ziliani in 2005 to improve scalable coding efficiency. However,
the LPU still needs to change the low-pass subband LP coefficients
due to the spatial update step in the decomposition procedure.
Hence, it has the same problem as the aforementioned lifted pyramid
method. The present invention solves the long felt needs of the
prior art attempts and presents novel methods that offer a variety
of unanticipated benefits.
Accordingly, it is desirable to provide advanced methods for
resampling and reconstruction within the pyramid representation
framework for digital signals. Such signals may be contaminated by
noise, either from quantization as in compression applications,
from transmission errors as in communications applications, or from
display-resolution limit adaptation as in multi-rate signal
conversion. The methods of the present invention offer enhanced
reconstruction.
SUMMARY OF THE INVENTION
The foregoing needs are met, to a great extent, by the present
invention, wherein in one aspect an apparatus is provided that in
some embodiments provide advanced methods for resampling and
reconstruction within the pyramid representation framework for
digital signals.
In accordance with one embodiment of the present invention, an
optimal laplace pyramid processing system is presented herein for
processing digital signal elements selected from a set of
dimensions within a signal, comprising a laplace pyramid
decomposition stage, and intermediate stage, and a laplacian
pyramid reconstruction stage. The laplace pyramid decomposition
stage includes a decimation having a signal as an input and a
coarse approximation of the signal as an output, and an
interpolation having the coarse approximation as an input and a
detail signal as an output. The laplacian pyramid reconstruction
stage has the coarse approximation and detail signal as inputs and
a reconstructed signal as an output, wherein the decimation retains
maximum energy in the coarse approximation and the reconstructed
signal is simultaneously a minimum mean square error approximation
of the original signal.
In accordance with another embodiment of the present invention, An
enhanced reconstruction laplacian pyramid processing system for
processing a plurality digital signal elements selected from any
set of dimensions within at least one signal, comprising a
laplacian pyramid decomposition stage, an intermediate stage, a
laplacian pyramid reconstruction stage, and an enhanced
reconstruction stage. The laplacian pyramid decomposition stage
includes a decimation having a signal as an input and a coarse
approximation of the signal as an output, and an interpolation
having the coarse approximation as an input and a detail signal as
an output. The laplacian pyramid reconstruction stage has the
coarse approximation and detail signal as inputs, and a
reconstructed signal as an output. The enhanced reconstruction
stage has the coarse approximation and reconstructed signal as
inputs and an enhanced reconstructed signal as an output.
There has thus been outlined, rather broadly, certain embodiments
of the invention in order that the detailed description thereof
herein may be better understood, and in order that the present
contribution to the art may be better appreciated. There are, of
course, additional embodiments of the invention that will be
described below and which will form the subject matter of the
claims appended hereto.
In this respect, before explaining at least one embodiment of the
invention in detail, it is to be understood that the invention is
not limited in its application to the details of construction and
to the arrangements of the components set forth in the following
description or illustrated in the drawings. The invention is
capable of embodiments in addition to those described and of being
practiced and carried out in various ways. Also, it is to be
understood that the phraseology and terminology employed herein, as
well as the abstract, are for the purpose of description and should
not be regarded as limiting.
As such, those skilled in the art will appreciate that the
conception upon which this disclosure is based may readily be
utilized as a basis for the designing of other structures, methods
and systems for carrying out the several purposes of the present
invention. It is important, therefore, that the claims be regarded
as including such equivalent constructions insofar as they do not
depart from the spirit and scope of the present invention.
BRIEF DESCRIPTION OF THE DRAWINGS
FIG. 1 is a perspective view illustrating a prior art multi-scale
Laplacian Pyramid (LP) signal representation.
FIG. 2 is a diagrammatic representation of a basic block diagram of
a prior art Laplacian Pyramid (LP) signal decomposition scheme.
FIG. 3 depicts a prior art frame-based pyramid reconstruction
scheme.
FIG. 4 illustrates a master pyramid reconstruction scheme in
accordance with one embodiment of the method of the present
invention.
FIG. 5 illustrates the reduction properties of the reconstruction
scheme of the present invention in relationship to that of the
prior art.
FIG. 6 depicts a master reconstruction scheme in accordance with
the present invention.
FIG. 7 shows the comparison of various reconstruction schemes from
the quantized LP coefficients of the popular 512.times.512 Barbara
test image using the reconstruction schemes of FIG. 1, FIG. 2, and
that of the present invention.
FIG. 8 illustrates the frequency responses of the equivalent
iterated filters for the three-level LP representation with the
low-pass filter h[n] in SVC via the conventional reconstruction
method shown in FIG. 1.
FIG. 9 depicts the frequency responses of the equivalent iterated
filters for the three-level LP representation with the low-pass
filter h[n] in SVC via the lifting-based reconstruction method in
accordance with the present invention.
FIG. 10A is a photographic illustration of a depiction of image
de-noising involving a first pyramid reconstruction scheme, as
compared with FIGS. 10B and 10C.
FIG. 10B is a photographic illustration of a depiction of image
de-noising involving a second pyramid reconstruction scheme, as
compared with FIGS. 10A and 10C.
FIG. 10C is a photographic illustration of a depiction of image
de-noising involving a third pyramid reconstruction scheme, as
compared with FIGS. 10A and 10B.
FIG. 11A is a photograph of a portion of the reconstructed Barbara
test image with severe aliasing effects using prior art
filters.
FIG. 11B is a photograph of a portion of a visually-pleasant
aliasing-free reconstructed Barbara test image in accordance with
the 13-tap SVC low-pass filter implemented in the present
invention.
FIG. 12 is a detail view of a comparison of the frequency responses
of the 9-tap, 7-tap, and 13-tap low-pass filters used in accordance
with the present invention.
FIG. 13 shows the frequency responses of a plurality of
down-sampling low-pass filters in accordance with the present
invention.
FIG. 14 shows the frequency responses of a plurality of up-sampling
low-pass filters in accordance with the present invention.
FIG. 15 shows the filter taps of a plurality of down-sampling
low-pass filters in accordance with the present invention.
FIG. 16 shows the filter taps of a plurality of up-sampling
low-pass filters in accordance with the present invention.
FIG. 17 presents a 13-tap low-pass filter in SVC and its
coefficients.
FIG. 18 presents a comparison of de-noising performances of various
LP reconstruction schemes.
DETAILED DESCRIPTION
The invention will now be described with reference to the drawing
figures, in which like reference numerals refer to like parts
throughout. An embodiment in accordance with the present invention
provides novel resampling filters and lifting-based techniques to
significantly enhance both the conventional LP decomposition and
reconstruction frameworks. The present invention embodies a
complete parameterization of all synthesis reconstruction schemes,
among which the conventional LP reconstruction and the frame-based
prior art pyramid reconstruction scheme are but special cases.
FIG. 1 illustrates a prior art method wherein a non-linear operator
is associated with each level of decomposition. This illustration
depicts the conventional multi-scale Laplacian Pyramid (LP) signal
representation 10 where the input signal x[n] 12 is represented as
a combination of a coarse approximation and multiple levels of
detail signals 14 and 16 at different resolutions. Optimal
reconstruction of the input signal x[n] 12 is consistent as the
reconstruction stage adds back what was subtracted during the
decomposition stage.
FIG. 2 depicts the basic block diagram of the Laplacian Pyramid
(LP) signal decomposition scheme 18. On the left is the LP analysis
stage which generates the coarse approximation signal c[n] 20 and
the prediction error or residue (details) signal d[n] 22. On the
right is the conventional LP synthesis stage where the signal x[n]
12' is reconstructed by combining the residue with the interpolated
coarse approximation.
For an LP with decimation factor M, the synthesis FB of the present
invention covers all the design space, but has only M design
parameters. This is in contrast to M(M+1) free entries in the
generic synthesis form presented in the prior art frame pyramid by
Do and Vetterli. The present invention leads to considerable
simplification in the design of the optimal reconstruction stage.
The dual frame reconstruction is also derived from the lifting
representations set forth in the present invention. The novel
reconstruction is able to control efficiently the quantization
noise energy in the reconstruction, but does not require
bi-orthogonal filters as they would otherwise be used in the prior
frame-based pyramid reconstruction.
A special lifting-based LP reconstruction scheme is also derived
from the present invention's master LP reconstruction, which allows
one to choose the low-pass filters to suppress aliasing in the low
resolution images efficiently. At the same time, it provides
improvements over the usual LP method for reconstruction in the
presence of noise. Furthermore, even in the classic LP context, the
resampling filters in accordance with the present invention are
optimized to offer the fewest mean squared reconstruction errors
when the detail signals are missing. In other words, with only the
lower-resolution coarse approximation of the signal available, the
present invention's pair of decimation and interpolation filters
deliver the minimum mean-squared error reconstruction while
capturing the maximum energy in the coarse signal. Furthermore, all
decimation and interpolation filter pairs are designed to be
hardware-friendly in that they have short finite impulse responses
(FIR), linear phase, and dyadic-rational coefficients.
FIG. 3 depicts the frame-based pyramid reconstruction scheme 24 as
described in Do and Vetterli's "Frame Pyramid." Operators H(z) and
G(z) in this method must be bi-orthogonal, i.e., their inner
product yields unity <h[n],g[n]>=1.
LPs are in one-to-one correspondence with pairs of up and down
sampling filters. Although such "resampling" filters are well-known
and commonly used, the present invention presents special up and
down sampling filters and corresponding LPs which display certain
optimization characteristics. Systems that employ them are
designated herein as Optimal Laplace Pyramid Processing Systems
(OLaPPS). For an LP to be qualified as an OLaPPS, it must exhibit
two main characteristics. First, the Decimation Filter H(z) has to
retain the maximum signal energy in the principal component sense.
In other words, the coarse approximation c[n] in an OLaPPS contains
at least as much signal energy as other approximation signals
obtained from other decimation filters. Second, the Interpolation
Filter G(z) yields a reconstructed signal {circumflex over (x)}[n]
that is optimal in the mean-squared sense. In other words,
{circumflex over (x)}[n] is the minimum mean-squared error
reconstruction of x[n] among available reconstructions.
An embodiment of the present inventive reconstruction method is
illustrated in FIG. 4. FIG. 4 shows the master pyramid
reconstruction scheme 32 as covered in this invention. Operators
employed in the analysis pyramid stage {H(z), G(z)} do not have to
satisfy the bi-orthogonal property or any other, as the
reconstruction method of the present invention leads to perfect LP
reconstruction for any operator P(z). This framework is a
significant improvement of the conventional LP reconstruction (the
first step in the reconstruction involving G(z)) and of the
frame-based pyramid LP reconstruction.
The filters of this embodiment of the present invention have roots
from the wavelet theory, which is well known in the art to have
excellent interpolation characteristics. The novel system of the
present invention ensures that if the re-sampled lower-resolution
signal ever has to be interpolated back to the original high
resolution, then the difference between the original
high-resolution signal and the reconstruction is minimized.
Moreover, the present invention demonstrates that efficiency of the
re-sampling system above does not necessarily have to be sacrificed
by employing short low-complexity integer-coefficient filters. One
potential application is in high-definition (HD) and
standard-definition (SD) video conversion where this inventive
OLaPPS interpolation ensures that the video for HD display
up-converted from an OLaPPS-processed SD source achieves the
highest quality level in the mean-squared sense.
FIG. 5 demonstrates that the frame-based pyramid reconstruction
scheme 34 is just a particular solution in the master framework of
the present invention: if one chooses to employ a set of
bi-orthogonal filter pair {H(z), G(z)} and furthermore sets
P(z)=G(z), then the reconstruction method 36 of the present
invention on the left reduces down to the frame-based
reconstruction method 38 on the right.
FIG. 6 depicts the master pyramid reconstruction structure 40 of
the present this invention. Here, D 42 can be any decimation
operator (can be non-linear) which produces a coarse approximation
c[n]44 of the input signal x[n] 46 while I 48 can be any
interpolation operator (can be non-linear) which attempts to
construct a full-resolution signal 50 resembling x[n] from the
coarse approximation c[n]. The final reconstruction step involves P
52 which can be any prediction operator (again, can be either
linear or non-linear).
FIG. 7 shows the comparison of various reconstruction schemes from
the quantized LP coefficients of the popular 512.times.512 Barbara
test image. In the graph 54 of quantization step size v. SNR, the
LP is decomposed with two levels; both H(z) and G(z) filters are
set as the low-pass filter employed in SVC
h[n]={2,0,-4,-3,5,19,26,19,5,-3,-4,0,2}. REC-1 56 is the result
from the traditional pyramid reconstruction in FIG. 1. REC-2 58
denotes the result of the prior art frame-based reconstruction
scheme proposed by Do and Vetterli and shown in FIG. 2. Finally,
REC-3 60 is the result from our reconstruction method where all
three filters (including the arbitrary operator P(z)) are set to
the filter H(z) above.
FIG. 8 illustrates the graph 62 of frequency responses of the
equivalent iterated filters for the three-level LP representation
with the low-pass filter h[n] in SVC [3] via the prior art
reconstruction method shown in FIG. 1. It is to be noted that all
synthesis filters are low-pass. FIG. 9 depicts the graph 64 of
frequency responses of the equivalent iterated filters for the
three-level LP representation with the low-pass filter h[n] in SVC
via the proposed lifting-based reconstruction method. Here, the
synthesis filters are band-pass and match with the frequency
regions of corresponding sub-bands. Therefore, the new method
confines the influence of noise from the LP only in these localized
sub-bands.
FIGS. 10A-10C illustrate a comparison in image de-noising involving
three pyramid reconstruction schemes. The Barbara test image is
corrupted with uniform independent identically distributed (i.i.d.)
noise introduced to 6-level decomposition LP coefficients with
13-tap low-pass filter in SVC. Conventional REC-1 reconstruction:
SNR (signal-to-noise ratio)=6.25 dB; Framed-based REC-2
reconstruction: SNR=14.17 dB; the general REC-3 reconstruction in
this invention: SNR=17.20 dB. This is a dramatic enhancement of the
image reconstruction.
FIGS. 11A and 11B demonstrates visually the importance of low-pass
filter design in our reconstruction approach. On the left is a
portion of the reconstructed Barbara test image with severe
aliasing effects where the two operators H(z) and G(z) are chosen
as the famous Daubechies 9/7-tap bi-orthogonal wavelet filters
(JPEG2000 default filter pair) respectively. On the right is a
portion of the visually-pleasant aliasing-free reconstructed
Barbara test image where the three operators H(z), G(z), and also
P(z) are all set to the 13-tap SVC low-pass filter [3].
Down-Sampling Odd-Length Filter Design
Instead of optimizing the low-pass filter so that its frequency
response has steep transition characteristics to match the ideal
low-pass box filter, implementation of the present invention calls
for a smoother, slower-decaying frequency response. Filters that
allow a little aliasing (to capture a bit more image information)
outperform filters with good anti-aliasing characteristics;
accordingly good wavelet filters tend to perform well here.
Therefore, three solution-based aspects of this embodiment of the
present invention are set forth herein: h5=[-1 2 6 2 -1]/8: 5-tap
dyadic-coefficient filter as used by JPEG2000. h9=[1 -1 -3 9 20 9
-3 -1 1]/32: 9-tap dyadic-coefficient filter, an improvement of the
default 9-tap irrational-coefficient Daubechies wavelet filter in
JPEG2000; h11=[1 0 -3 0 10 16 10 0 -3 0 1]/32: 11-tap
dyadic-coefficient half-band filter designed to minimize aliasing
effects in sub-sampled images.
FIG. 12 compares the frequency responses of the 9-tap, 7-tap, and
13-tap low-pass filters used for demonstration frequently in the
description. FIG. 13 shows the frequency responses of several of
the down-sampling low-pass filters. FIG. 15 presents a table of
dyadic-rational coefficients or elements of decimation filters.
Down-Sampling Even-Length Filter Design
Following a similar design philosophy as with the odd-length
filters in the previous section, the down-sampling even length
filter design of the present invention presents maxflat half band
filters and performs spectral factorization to obtain even-length
filter pairs for down- and up-sampling. This design procedure
ensures that each filter pair forms a pair of bi-orthogonal
partners, minimizing the mean-square error of the reconstruction
signal. Accordingly, two solution-based aspects of this embodiment
of the present invention are set forth herein: h4=[-1 3 3 -1]/4:
4-tap dyadic-coefficient filter; h8=[3 -9 -7 45 45 -7 -9 3]/64:
8-tap dyadic-coefficient filter.
The frequency responses of several of the proposed filters,
even-length as well as odd-length, are depicted in FIG. 13 along
with the previous H.264 low-pass filter's response. Besides FIR and
integer coefficients, all of the filters have linear phase, a
critical requirement for imaging applications and fast
implementation. All of the decimation filters are tabulated in FIG.
15.
Up-Sampling Filter Design
Filters with good anti-aliasing characteristics and smooth
frequency responses (a characteristic of maximally-flat or maxflat
filters for short [9, 10, 13]) perform well in up-sampling. The
prior art 11-tap filter in H.264 SVC has both of these properties.
The present invention provides another 7-tap candidate with similar
characteristics and performance level, yet requiring a much lower
computational complexity: f7=[-1 0 9 16 9 0 -1]/16. The odd-length
filter pair of h9/f7 is designed from approximations of wavelet's
famous 9/7 Daubechies filters used as the default choice in
JPEG2000, which in turn are obtained from spectral factorization of
the maxflat half-band filter p15=[-5 0 49 0 -245 0 1225 2048 1225 0
-245 0 49 0 -5]/2048.
For the shorter even-length pairs of h4/f4 and h8/f4, we start with
the following two shorter maxflat half-band filters: p7=[-1 0 9 16
9 0 -1]/16 p11=[3 0 -25 0 150 256 150 0 -25 0 3]/256. The
even-length anti-imaging up-sampling filter is chosen as f4=[1 3 3
1]/4 and the remaining roots of p7 and p11 are allocated to h4 and
h8 respectively. The frequency responses of all up-sampling filters
as well as of the previous 11-tap H.264 filter are shown in FIG.
14. The solutions of the present invention sacrifice sharp
frequency transition for a higher degree of smoothness/regularity.
This is a desirable characteristic for smooth interpolation. All of
the FIR linear-phase integer-coefficient interpolation filters of
the present invention are tabulated in FIG. 16. Laplacian Pyramids
as Oversampled Filter Banks
The prior art LP decomposition and its usual reconstruction can be
illustrated in FIG. 2, where H(z) and G(z) are the decimation and
interpolation filters, respectively. In the LP decomposition, the
coarse approximation c[n] of an input signal x[n] is generated
through the H(z) filtering stage followed by down-sampling. Then,
c[n] is up-sampled and filtered to provide a prediction signal
whose difference from the original signal x[n] is called the
prediction error signal d[n]; this typically contains
high-frequency finer details of x[n]. In the conventional LP
reconstruction, the reconstruction signal {circumflex over (x)}[n]
is obtained by simply adding d[n] back to the interpolation of
c[n]. Since c[n] and d[n] have more coefficients than x[n], the LP
is an over-complete system, often called a frame or an over-sampled
filter bank in the literature.
The LP realizes a frame expansion, as x[n] can be always
reconstructed from c[n] and d[n]. From the Filter Bank (FB) point
of view, the LP can be formulated as an (M+1)-channel over-sampled
FB with a sampling factor M [4]. Let the superscript letter H
denote the Hermitian transpose, then the polyphase analysis matrix
for the LP decomposition in FIG. 2 can always be written as
.function..function..function..times..function. ##EQU00001## where
the 1.times.M vectors h(z) and g(z) are Type-I polyphase matrices
of H(z) and G(z), respectively [13]. The corresponding polyphase
synthesis matrix is
.function..function..times..times. ##EQU00002##
It can be easily shown that perfect reconstruction is always
achieved in the absence of noise regardless of the selection of
H(z) and G(z), since the cascade of the analysis followed by the
synthesis polyphase matrices is always the identity matrix, i.e.,
R(z) E(z)=I.
As illustrated in FIG. 3, the prior art frame-based LP
reconstruction scheme of Do and Vetterli has the polyphase
synthesis matrix as
.function..function..function..times..function. ##EQU00003## The PR
condition is satisfied only when H(z) and G(z) are bi-orthogonal
filters, and the reconstruction above leads to an improvement over
the traditional reconstruction when H(z) and G(z) are orthogonal or
near orthogonal filters. Under this restriction, E(z) is a
paraunitary matrix.
Lifting-based constructions are utilized extensively in U.S. Pat.
No. 6,421,464, "Fast Lapped Image Transforms Using Lifting Steps,"
by the inventors of the present invention. For example, in the
elementary two-dimensional case, a lifting step corresponds to a
2.times.2 matrix that is the identity plus one non-diagonal entry,
and whose inverse is the same matrix, but the non-diagonal entry
has the opposite sign. Lifting steps are ideal for constructing and
implementing highly optimized signal transforms. They are used here
for optimized integer-based resampling filters and associated
LPs.
A second embodiment of the present invention pertains to enhanced
reconstruction methods, applicable even when the resampling filters
are fixed. For any given LP filters H(z) and G(z), the PR condition
can be always satisfied, since by construction the error signal is
incorporated into the scheme. In the prior art scheme of Do and
Vetterli, a general complete parameterization of all PR synthesis
FBs is formulated as
.function..function..function..function..function..times..function.
##EQU00004## where {tilde over (R)}(z) can be any particular left
inverse of E(z), and U(z) is an M.times.(M+1) matrix with bounded
entries. The reconstruction scheme resulting from equation (4) thus
has M(M+1) degrees of design freedom. In this second embodiment of
the present invention, the number of free parameters can be further
reduced based on the following lifting-based parameterization.
For any LP filters, the polyphase matrix in Eq. (1) can always be
factorized into two lifting steps as follows
.function..times..function..function..times..function..function..times..-
function..times. ##EQU00005## To invert a lifting step, one can
subtract out what was added in at the forward transform. Thus, the
left inverse of E(z) is achieved by inverting the lifting steps in
Eq. (5). This provides the master form of R(z).
For any given conventional LP analysis (decomposition) stage, its
synthesis polyphase matrix R(z) has the following master
lifting-based representation, is hereby designated as an Enhanced
Reconstruction Laplace Pyramid (ERLaP):
.function..function..function..times..function..times..function.
##EQU00006## where p(z) is any arbitrary 1.times.M vector with
bounded entries. The first two terms in the matrix product in Eq.
(5) are lower-triangular and upper-triangular square matrices, so
it is easy to see that their corresponding inverses are similar
triangular matrices with inverting polarity as in the last two
terms in the matrix product of Eq. (6). What remains is to obtain
the left inverse for the (M+1).times.M matrix
##EQU00007## which has a row of M zeros on top of an identity
matrix. The most general left inverse of this matrix is [p.sup.H
(z) I.sub.M] where p(z) is an arbitrary polynomial vector taking
the form described above and the superscript H indicates the
conjugate transpose operator since
.function..function. ##EQU00008## Finally, the matrix [p.sup.H (z)
I.sub.M] can always be factorized into the following product
.function..function. ##EQU00009## as shown in the first two terms
of Eq. (6).
Let p(z) be the type-I polyphase vector of a filter P(z). Then, the
reconstruction matrix in Eq. (6) is equivalent to the master
reconstruction scheme of the first embodiment of the present
invention shown in FIG. 4. For any given LP decomposition, Eq. (6)
only has M degrees of design freedom. Despite the reduced number of
free parameters, Eq. (6) covers the complete space of all synthesis
filter banks. It is to be noted that the operator P(z) is
independent of the decimation filter H(z) and the interpolation
filter G(z). How to optimize P(z) for any given pair of H(z) and
G(z) is the topic of interest in the next section. From Eq. (6) and
the equivalent representation in FIG. 4, given that the first
reconstruction stage involving G(z) incorporates the conventional
pyramidal reconstruction, the second embodiment of the present
invention groups the two stages involving H(z) and G(z) into a
combined operator called the Enhanced Reconstruction stage. The
conventional pyramidal reconstruction stage and the Enhanced
Reconstruction stage forms an ERLaP as first described in Eq.
(6).
Dual-Frame LP Reconstruction Scheme and Optimal Design
For any filters H(z) and G(z), the reconstruction synthesis matrix
as shown in Eq. (6) can have certain desired properties by
optimizing p(z). In order to choose p(z) such that Eq. (6)
minimizes the reconstruction error when white noise is introduced
into LP coefficients, the optimization solution presented herein is
to find the dual frame reconstruction solution. Through error
analysis of the LP system, a close-form solution of dual frame
reconstruction is presented below.
For the LP with polyphase analysis matrix E(z) given in Eq. (1),
its dual frame reconstruction can be expressed as
.function..function..times..function..times..function..function..function-
. ##EQU00010## where
.function..function..function..times..function..function..times..function-
..function..times..function. ##EQU00011##
and d(z)=1-h(z)g.sup.H(z). (9)
It is to be noted that once given FIR filters H(z) and G(z), the
dual frame solution above corresponds to a FB with infinite-impulse
response (IIR) filters. If L(z)=d(z)d.sup.H(z)+h(z)h.sup.H(z) is a
positive constant, then the dual-frame solution is a FB with FIR
filters. Otherwise, L(z) is approximated by a constant to realize
an FIR implementation.
Considering the dual frame reconstruction in Eq. (7) that normally
involves IIR filters and hence is undesirable in practical
applications, a second aspect of the second embodiment of the
present invention of the master lifting-based LP reconstruction in
Eq. (6) and let p(z)=g(z). This special LP reconstruction then
leads to the LP reconstruction scheme depicted in FIG. 5. Recall
that when H(z) and G(z) are not bi-orthogonal filters, the prior
art frame-based pyramid reconstruction of Do and Vetterli does not
satisfy the PR condition. Thus, its performance would suffer.
However, the LP reconstruction of the present invention always
satisfies the PR condition regardless of filter choices, and it can
still maintain good performance when H(z) and G(z) are reasonable
low-pass filters. As an illustration, let REC-1 denote the usual
reconstruction shown in FIG. 2, REC-2 denote the prior art
frame-based pyramid reconstruction of Do and Vetterli depicted in
FIG. 3, and REC-3 denote the special lifting reconstruction
illustrated in FIG. 5. The performance of these three LP
reconstruction schemes is compared when H(z) and G(z) are the same
low-pass filter in SVC whose coefficients are tabulated in FIG. 17
and when M=2. FIG. 17 presents a 13-tap low-pass filter in SVC and
its coefficients.
First, an image coding application is used wherein uniform scalar
quantization with equal step size is applied for all LP
coefficients (in an open-loop mode). FIG. 7 shows the SNR result
for the popular Barbara test image of size 512.times.512. It
demonstrates that REC-3 has 0.5 dB gain over REC-1, while REC-2 is
around 2.5 dB worse than REC-1. Secondly, a prior art de-noising
application is used wherein the LP coefficients are usually
thresholded so that only the m most significant coefficients are
retained. FIG. 18 lists the numerical de-noising results for three
standard test images. REC-3 consistently yields better performances
by around 0.4 dB in SNR than REC-1 while REC-2 has worse
performance than REC-1 since the PR property is not satisfied. It
is to be noted that when the LP filters are bi-orthogonal, e.g.,
9/7 bi-orthogonal wavelet filters, REC-3 has exactly the same
performance as REC-2, which can provide better performance than
REC-1 by around 0.5 dB in SNR as presented in the prior art.
However, bi-orthogonal filters could introduce annoying aliasing
components into low-resolution LP subbands, especially in image
texture and/or edges regions, while the low-pass filter can
generate more pleasing visual quality.
The multilevel representation is achieved when the LP scheme is
iterated on the coarse signal c[n]. For the prior art LP
reconstruction in FIG. 2, FIG. 8 shows an example of frequency
responses of the equivalent filters when the LP filters are the
low-pass filter from FIG. 17. It depicts that the synthesis filters
from the conventional LP reconstruction scheme are either low-pass
or all-pass filters. On the other hand, FIG. 9 illustrates the
frequency responses of the equivalent filters of the second
embodiment of the present invention's master reconstruction scheme
in FIG. 5. It can be observed that the filters here are now
band-pass and match with the frequency response regions of
corresponding sub-bands. Thus, the inventive REC-3 reconstruction
scheme can confine the errors from high-pass sub-bands of a
multi-level LP decomposition.
This leads to better performance than REC-1 in coding applications.
It also has the prominent advantage over REC-1 when the errors in
the LP coefficients have non-zero mean. In such case, with the
REC-1 reconstruction, the nonzero mean propagates through all
low-pass synthesis filters and appears in the reconstructed signal.
On the contrary, with REC-3 reconstruction, the nonzero mean is
cancelled by the band-pass filters. Herein, the same examples are
used as presented in the prior art: the errors in the LP
coefficients (6 levels of LP decomposition) are uniformly
distributed in [0, 0.1]. The SNR values for three reconstruction
schemes REC-1, REC-2, and REC-3 are 6.25 dB, 14.17 dB and 17.20 dB,
respectively. Although the synthesis functions of REC-3 have
similar frequency responses to those of REC-2, the inventive
reconstruction scheme of the present invention has better noise
elimination performance because REC-2 does not satisfy the PR
condition for the given low-pass filter.
Although an example of the system is shown relative to image and
video data, it will be appreciated that the system may also be
applied to the processing of uni- and multi-dimensional discrete
signals such as audio, radar, sonar, natural images, photographs,
drawings, multi-spectral images, volumetric medical image data
sets, and video sequences, etc, at multiple resolutions that are
captured directly in digital format or after they have been
converted to or expressed in digital format.
The many features and advantages of the invention are apparent from
the detailed specification, and thus, it is intended by the
appended claims to cover all such features and advantages of the
invention which fall within the true spirit and scope of the
invention. Further, since numerous modifications and variations
will readily occur to those skilled in the art, it is not desired
to limit the invention to the exact construction and operation
illustrated and described, and accordingly, all suitable
modifications and equivalents may be resorted to, falling within
the scope of the invention.
* * * * *